#758222

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

5492 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

(6,8)(9,10,13,12)(11,15,16,14)(17,18)(19,20)(21,23)(22,24);(2,4)(6,8)(17,19,21,24)(18,22,23,20);(1,2,3,4)(5,6,7,8)(19,24)(20,22);(9,11,13,16)(10,14,12,15)(17,20)(18,19)(21,22)(23,24);(9,12,13,10)(11,15,16,14)(17,18)(19,20)(21,23)(22,24)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fh,d^2=i,e^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72333975, #72333989, #72334026
maximal normal subgroup of	#72333975, #72333989, #72334026

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 665418);